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Properties

Label 2-115920-1.1-c1-0-102
Degree $2$
Conductor $115920$
Sign $-1$
Analytic cond. $925.625$
Root an. cond. $30.4240$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s + 4·11-s − 6·13-s + 2·17-s − 23-s + 25-s − 6·29-s + 35-s + 2·37-s − 6·41-s − 4·43-s + 49-s − 6·53-s + 4·55-s − 12·59-s + 10·61-s − 6·65-s − 4·67-s + 4·71-s − 2·73-s + 4·77-s + 2·85-s + 10·89-s − 6·91-s − 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.20·11-s − 1.66·13-s + 0.485·17-s − 0.208·23-s + 1/5·25-s − 1.11·29-s + 0.169·35-s + 0.328·37-s − 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.824·53-s + 0.539·55-s − 1.56·59-s + 1.28·61-s − 0.744·65-s − 0.488·67-s + 0.474·71-s − 0.234·73-s + 0.455·77-s + 0.216·85-s + 1.05·89-s − 0.628·91-s − 0.203·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115920\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(925.625\)
Root analytic conductor: \(30.4240\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.01408537281847, −13.33061919405445, −12.87575145420780, −12.32869750831010, −11.82425651230234, −11.64948308426184, −10.91015947428211, −10.39006804571418, −9.837924172815868, −9.446831260364853, −9.134667712221865, −8.402874028741411, −7.872662941931274, −7.359576450775543, −6.888791189254538, −6.358114114121154, −5.755955214236100, −5.208702858568307, −4.712570128861239, −4.194305838837878, −3.449195476966121, −2.956754505810014, −2.015517978113117, −1.803176480548426, −0.9133045823435184, 0, 0.9133045823435184, 1.803176480548426, 2.015517978113117, 2.956754505810014, 3.449195476966121, 4.194305838837878, 4.712570128861239, 5.208702858568307, 5.755955214236100, 6.358114114121154, 6.888791189254538, 7.359576450775543, 7.872662941931274, 8.402874028741411, 9.134667712221865, 9.446831260364853, 9.837924172815868, 10.39006804571418, 10.91015947428211, 11.64948308426184, 11.82425651230234, 12.32869750831010, 12.87575145420780, 13.33061919405445, 14.01408537281847

Graph of the $Z$-function along the critical line